Intermittent electron density and temperature fluctuations and associated fluxes in the Alcator C-Mod scrape-off layer

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Intermittent electron density and temperature fluctuations and associated fluxes in the Alcator C-Mod scrape-off layer

R. Kube,^∗ O. E. Garcia, and A. Theodorsen Department of Physics and Technology,

UiT The Arctic University of Norway, N-9037 Tromsø, Norway

D. Brunner, A. Q. Kuang, B. LaBombard, and J. L. Terry MIT Plasma Science and Fusion Center, Cambridge, MA, 02139, USA

(Dated: March 16, 2018)

Abstract

The Alcator C-Mod mirror Langmuir probe system has been used to sample data time series of fluctuating plasma parameters in the outboard mid-plane far scrape-off layer. We present a statistical analysis of one second long time series of electron density, temperature, radial electric drift velocity and the corresponding particle and electron heat fluxes. These are sampled during stationary plasma conditions in an ohmically heated, lower single null diverted discharge. The electron density and temperature are strongly correlated and feature fluctuation statistics similar to the ion saturation current. Both electron density and temperature time series are dominated by intermittent, large-amplitude burst with an exponential distribution of both burst amplitudes and waiting times between them. The characteristic time scale of the large-amplitude bursts is approximately 15µs. Large-amplitude velocity fluctuations feature a slightly faster characteristic time scale and appear at a faster rate than electron density and temperature fluctuations. Describ- ing these time series as a superposition of uncorrelated exponential pulses, we find that probability distribution functions, power spectral densities as well as auto-correlation functions of the data time series agree well with predictions from the stochastic model. The electron particle and heat fluxes present large-amplitude fluctuations. For this low-density plasma, the radial electron heat flux is dominated by convection, that is, correlations of fluctuations in the electron density and radial velocity. Hot and dense blobs contribute approximately 6% of the total fluctuation driven heat flux.

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I. INTRODUCTION

Turbulent flows in the scrape-off layer (SOL) of magnetically confined plasmas have received great attention recently. Experimental analyses have demonstrated that plasma blobs propagating through the scrape-off layer towards the vessel wall dominate the plasma particle and heat fluxes at the outboard mid-plane [1–11]. In order to assess expected erosion and damage to the plasma enclosing vessel, the statistics of the impinging plasma fluxes are of great interest [12–15].

Plasma blobs are pressure perturbations spatially localized in the plane perpendicular to the magnetic field and elongated along the magnetic field lines. They are believed to be cre- ated in the vicinity of the last closed magnetic flux surface with particle density perturbation amplitudes comparable in magnitude to the average scrape-off layer particle density. At the outboard mid-plane location the magnetic curvature vector and field strength gradient point towards the magnetic axis. This causes an electric polarization of the blob structure due to magnetic curvature and gradient drifts. The resulting electric field propagates the blob towards the vessel wall, resulting in large cross-field particle and heat fluxes onto plasma facing components. [16–22].

Scrape-off layer plasma fluctuations furthermore exhibit several universal features. Time series data of plasma density fluctuations feature non-gaussian values of sample skewness and flatness and their probability density functions (PDFs) present elevated tails for large amplitude events. This feature has been observed in experiments [9–11, 23–29] as well as in numerical simulations [6, 30–33] and is well documented to be due to the radial propagation of plasma blobs [25, 34–40]. A quadratic relation between sample skewness and flatness has been reported from several experiments [26, 41–47]. Conditionally averaged waveforms of electron density time series exhibit approximately two-sided exponential waveform shapes [3–5, 25, 26, 46–49]. Several experiments report large-amplitude electron density fluctuations in phase with an outwards E×B drift velocity, that is radial particle flux events [3, 26, 46, 48, 50, and 51].

A recently developed stochastic model describes such time series as a super-position of uncorrelated pulses [52]. Assuming an exponential pulse shape and exponentially distributed pulse amplitudes and waiting times between pulses [25, 27, 29, 44, and 53] it predicts the fluctuation amplitudes to be Gamma distributed. The quadratic relation between moments

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of skewness and flatness of a Gamma distributed variable is in excellent agreement with the quadratic relation observed in experiments. This model furthermore predicts the ex- perimentally observed exponential density profiles in scrape-off layer plasmas [47 and 54].

The stochastic model has been generalized to describe general pulse shapes as well as additive noise. Analytic expressions of probability density functions, auto-correlation functions, power spectral densities and level crossing rates have been derived [27, 47, 55, and 56].

In this contribution it is demonstrated that the model predictions compare favorably with measurements of the fluctuating electron density and temperature, as well as with the radial velocity. It should be noted that by constructing the stochastic model as a superposition of individual pulses, the underlying non-linear dynamics of the plasma is parameterized.

Specifically, the steepening of radially propagating blob structures is modeled by exponential pulse shapes. Another approach, which proposes a stochastic differential equations to describe the non-linear plasma dynamics, under the constraint that the fluctuations are Gamma distributed, has recently been explored [57].

Scrape-off layer plasmas are usually diagnosed with Langmuir probes. They allow for three fundamental modes of operation. One way is to apply a sweeping voltage to a Lang- muir electrode. This allows the plasma density, the electric potential as well as the electron temperature to be inferred on the time scale of the sweeping voltage. This time scale is commonly of the orders of milliseconds, as to avoid hysteresis effects which arise at higher sweeping frequencies [58]. On the other hand, the time scale associated with blob propagation is on the order of microseconds. Conventional sweeping modes can thus not be used to investigate plasma fluctuations.

A second way is to bias the Langmuir electrode to a large negative electric potential relative to the vacuum vessel. This way the electrode draws the ion saturation current [59]

I_s= 1 2en_eA_p

rk_bT_e

m_i . (1)

Here e is the elementary charge, n_e is the electron density, T_e is the electron temperature, m_i is the ion mass and A_p is the current collecting probe area. This assumes that the ion temperature is zero, although it is typically larger than the electron temperature for the measurements in this paper. Employing a Reynolds decomposition of the time-dependent quantities in Eq. ( 1), as u(t) =u₀+u(t) for a variable u, shows that fluctuations ine n_e and

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T_e perturb the ion saturation current as Ie_s

I_s,0 ≈ en_e n_e,0 + 1

2 Te_e T_e,0

!

. (2)

Here the factor 1/2 comes from an expansion of the square-root for small relative electron temperature fluctuations. For equal relative fluctuations of the electron density and temperature the electron density contributes twice as much to the relative fluctuation of the ion saturation current than the electron temperature fluctuation. With no fast measurements of T_e at hand, a constant value is often assumed for T_e in Eq. ( 1) to find n_e given I_s.

A third mode of operation is to electrically isolate the Langmuir electrode. In this mode, it assumes the floating potential

V_f =V_p−ΛT_e, (3)

whereV_pis the plasma potential and Λ≈2−3 in scrape-off layer plasmas [60 and 61]. Using again a Reynolds decomposition, the fluctuating floating potential is given byVe_f =Ve_p−ΛTe_e. Thus, perturbations in the floating potential are equally due to fluctuations in the plasma potential and the electron temperature. Fast measurements ofT_e in the scrape-off layer are often unavailable such that n_e is approximated by I_s and the plasma potential is estimated by the floating potential. From this, an estimate of the radial E×B drift velocity can be calculated given two spatially separated measurements of the floating potential. It was recently observed that perturbations of the electron temperature may alter the estimated radial drift velocity [62–64].

Since plasma blobs present perturbations of the plasma density, temperature and electric potential, real time measurements of all three quantities from a single point are desirable as to precisely quantify their contributions to cross-field transport in the scrape-off layer.

Recent probe designs, such as ball pen probes [65 and 66] and emissive probes [67–70] allow fast sampling of the plasma potential but to evaluate the electron temperature one still needs to combine data from multiple electrodes.

Langmuir probe implementations that utilize multiple electrodes to provide real time samples of the fluctuating plasma parameters, such as triple probes, are routinely operated in several major tokamaks [71–74]. In this configuration current and voltage samples from different Langmuir electrodes are combined as to estimate the fluctuating electron density, the plasma potential, and the electron temperature in real time. On the other hand the

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equations of the triple probe configuration assume that the electrodes sample a homogeneous plasma. These assumptions are often violated in the scrape-off layer, where the characteristic length of the turbulence structures may be smaller than the separation of the Langmuir electrodes. Triple probe configurations have also been implemented in the time domain, removing the assumption of a homogeneous plasma [75–77]. This configuration requires two spatially separated Langmuir electrodes. Periodically biasing the electrodes to three different bias voltages allows to infer the electron density and temperature, as well as the plasma potential at each Langmuir electrode independently.

Fast measurements of electron temperature fluctuation in scrape-off layer plasmas are sparse. Measurements based on the method of harmonics [78] taken in the DIII-D tokamak suggest that fluctuations of the electron temperature and the electric drift velocity appear on average in phase with fluctuations in the electron density [2 and 4]. However, the method of harmonics has a time resolution of 10µs, comparable to the time scale of the turbulence structures in the plasma [78]. Analysis of an 8 ms long electron temperature data time series, taken by a triple probe configuration in the SINP tokamak, suggests that it presents the same non-gaussian features as commonly observed in electron density time series: the frequent arrival of large-amplitude bursts and heavy-tailed histograms [51]. Recent measurements reported from ASDEX Upgrade confirm that fluctuations of the electron density and temperature appear in phase, together with fluctuations in the plasma potential [62].

It was furthermore reported that the temperature fluctuations show on average a temporal asymmetry around the density peaks. Relative fluctuation levels of the electron temperature were found to be lower by a factor of approximately 2−3 than for the electron density.

The novel mirror Langmuir Probe (MLP) biasing technique allows for fast sampling of the ion saturation current, the electron temperature and the floating potential at a single sampling position [79 and 80]. This diagnostics consists of three major components. The actual mirror Langmuir probe is an electronic circuit that generates a current-voltage (I-V) characteristic with the three adjustable parameters I_s, T_e, and V_f:

I_MLP =I_s

exp

V −V_f T_e

−1

(4) The second main component is a Langmuir electrode immersed in the plasma to be sampled.

Both components are connected to a fast switching biasing waveform. The bias waveform switches between the states (V⁺, V⁰, V⁻), such that the Langmuir electrode draws approx-

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imately ±I_s at the states V^∓ and zero net current when biased to V⁰, as shown in Fig. 1 of [80]. Every 300 ns the bias voltage state is updated. Current samples from the MLP and the Langmuir electrode are compared after the bias voltage has settled. In order to minimize the deviation between the two sample pairs, the MLP adjusts the I_s, T_e, and V_f parameters dynamically. The main task of the MLP circuit is to set and maintain the optimal range of the bias voltages such that a complete I-V characteristic can be reconstructed from measurements at the Langmuir electrode at the three bias voltages states. Samples of the I-V response from the MLP and the Langmuir electrode are digitized at 3 MHz, synchronized to the states V⁺, V⁰, and V⁻. The current and voltage samples of the Langmuir electrode are then used for a fit to the I-V characteristic as to obtain T_e, V_f and I_s. The time stag- gering of the three sequential measurements voltages is neglected. Time series of the fit parameters at a sampling frequency of 1 MHz are obtained by mapping them one-to-one to the time samples of the voltage states V⁺, V⁰ and V⁻. Finally, the data time series of the T_e, V_f, and I_s fit parameters are linearly interpolated on the same time base with 3 MHz sampling frequency. From these sample values the electron density and the plasma potential are calculated [80].

This contribution presents a statistical analysis of exceptionally long data time series measured by the MLP in stationary plasma conditions. Section II describes the experimental setup and Sec. III describes the data analysis methods employed. The statistical properties of the ion saturation current, floating potential, as well as electron density and temperature are discussed in Sec. IV. Fluctuation time series of the radial velocity, the radial electron particle and heat fluxes are analyzed in Sec. V. A discussion and a conclusion of the results are given in Secs. VI and VII. Supplementary information on the stochastic model and on analysis of the MLP data is given in the Appendices A and B.

II. EXPERIMENTAL SETUP

Alcator C-Mod is a compact, high-field tokamak with major radius R = 0.68 m and minor radius a = 0.21 m [81–83]. It allows for an on-axis magnetic field strength of up to 8 T so as to confine plasmas with up to two atmospheres pressure. In this contribution we investigate the outboard mid-plane scrape-off layer of an ohmically heated plasma in a lower single-null diverted magnetic configuration with an on-axis toroidal field ofB_T = 5.4 T. The

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toroidal plasma current for the investigated discharge isI_p = 0.55 MA and the line averaged core plasma density is given by n_e/n_G = 0.12, where n_G is the Greenwald density. Such low density plasmas feature a far scrape-off layer with vanishing electron pressure gradients along magnetic field lines. The temperature drop from outboard mid-plane to the divertor plates is supported by the divertor sheaths [84].

A Mach probe head was dwelled at the limiter radius, approximately 0.11 m above the outboard mid-plane location. Its four electrodes are arranged in a pyramidal dome geometry on the probe head such that they sample approximately the same magnetic flux surface. Each electrode is connected to a MLP bias drive, and labeled northeast, southeast, southwest and northwest. Tracing a magnetic field line from the outboard mid-plane to the probe head, the east electrodes are in the shadow of the west electrodes, with the south electrodes facing the outboard mid-plane. The MLPs obtainI_s,V_f, andT_efrom fits to the I-V samples with a sampling rate of approximately 1 MHz. Further details on the probe head are given in [85].

III. DATA ANALYSIS

MLPs have been used successfully to measure profiles of average values and relative fluctuation levels [80]. However, large-amplitude fluctuations in the far scrape-off layer present challenges to interpreting the reported fit values I_s, V_f, and T_e. The MLP dynamically up- dates the voltage states V⁺ and V⁻ relative to a running average of electron temperature samples over a 2 ms window, V⁺−V⁻ < 4T_e holds, where T_e denotes this running average. When the instantaneous electron temperature at the Langmuir electrode significantly exceedsT_e, the range of the biasing voltages may be insufficient to resolve the I-V characteristic. This can result in large uncertainties of the fit parameters. Moreover, events unrelated to the turbulent plasma flows, such as probe arcing, may also produce in erroneous values of the fit values.

Parameters from I-V fits reported from all four MLPs at a given time were compared to investigate the robustness of the measured fluctuations. It was found that I_s,T_e, and V_f fit values reported from the four MLPs are of comparable magnitude when V⁺−V⁻ > 4T_e holds. On the other hand the fourT_e values may feature large outliers whenV⁺−V⁻ <4T_e holds. Therefore we analyze data time series obtained by applying a 12-point Gaussian filter on the current time samples obtained at the electrode biasing potentials (V⁺, V⁰, V⁻). The

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I_s, V_p and V_f time series used throughout this article are taken from the southwest MLP.

The used n_e and T_e time series are given by the average of the fit parameters reported by all four MLPs.

Figure 1 shows one millisecond long sub-records of the I_s, n_e, T_e, V_p and V_f data time series. Local maxima of the ion saturation current time series exceeding 2.5 times the sample root mean square value are marked with a red circle and 50µs long sub-records surrounding these local maxima are marked in black in all data time series. A visual inspection suggest that large amplitude fluctuations in the ion saturation current are correlated with similar large amplitude fluctuations in the electron density and temperature time series. These large-amplitude bursts appear to occur on a similar time scale. The Pearson sample correlation coefficient for the I_s and n_e time series is given by R_I_s_,n_e = 0.91. This substantiates the approach taken in the analysis of conventional Langmuir probe data time series, namely that fluctuations in I_s are used as a proxy for fluctuations in n_e. Furthermore, we find a sample correlation coefficient forI_s andT_e given byR_I_s_,T_e = 0.83. This suggests that fluctuation statistics are similar for these two time series. The plasma potential and the electron temperature present fluctuations on similar time scales. However, there is no correlation between large amplitude fluctuations apparent between the two time series. Fluctuations in the floating potential are anti-correlated to fluctuations in the ion saturation current, with a Pearson sample correlation coefficient given byR_I_s_,V_f =−0.33.

For further analysis of the data time series we rescale them as to have locally vanishing mean and unity variance:

Ψ =e Ψ− hΨi_mv

Ψ_rms,mv . (5)

The moving average and moving root mean square time series are computed from samples att_i =i4_t as

hΨi_mv(t_i) = 1 2r+ 1

Xr k=−r

Ψ(t_i+k), (6)

Ψ_rms,mv(t_i) =

"

1 2r+ 1

Xr k=−r

(Ψ(t_i+k)− hΨ(t_i)i_mv)²

#1/2

. (7)

where4_t= 0.3µs is the sampling time. Using a filter radiusr= 16384, which corresponds to approximately 5 ms, ensures that both the moving average and the moving root mean square time series feature little variation. Indeed, the sample averages of all rescaled time series

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0.025 0.050

Is/A

5 10

ne/1018m−3

10 15 Te/eV

20 40

Vp/V

0.00 0.25 0.50 0.75 1.00

time/ms

−20 0 Vf/V

FIG. 1. Time series of the ion saturation current, electron density and temperature, and plasma and the floating potentials. Local maxima exceeding 2.5 times the root mean square value of the I_s time series are marked by red dots. The black lines mark 50µs long sub-records centered around these maxima.

are approximately 10⁻³ and their standard deviations deviates from unity by a comparable factor.

Figure 2 illustrates this rescaling. It shows the T_e time series in physical units in green.

The moving average, defined by Eq. ( 6), is shown by the solid black line and flanked by the moving root mean square, shown by the dashed black lines. While the moving root mean square varies little, between 2.5 and 3.5 eV, the moving sample average varies between 12 and 19 eV. Absorbing these variations into the normalization of the time series allows to compare samples of the entire one second long data time series.

All rescaled data time series present non-vanishing sample coefficients of skewness and excess kurtosis, or flatness, listed in Tab. I. While the electron density and temperature time series feature comparable coefficients of skewness, this moment is larger for the ion saturation current. Similarly, the flatness of the ion saturation current time series is consistently larger

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0.80 0.82 0.84 0.86 0.88 0.90 time/s

5 10 15 20 25 30

Te/eV

FIG. 2. The electron temperature time series (green) and its moving average, defined by Eq. ( 6) (black solid line). The black dashed lines are one moving standard deviation, defined by Eq. ( 7), above and below the moving average.

than for either electron quantity. The floating potential features negative coefficients of sample skewness and non-vanishing coefficients of flatness. On the other hand the plasma potential is skewed towards positive values and also features positive coefficients of sample kurtosis.

Quantity Skewness Flatness

Ie_s 1.1/1.0/1.2/1.1 2.0/1.7/2.5/2.1 Ve_f −0.23/−0.23/−0.83/−0.64 0.031/0.18/0.86/0.66 Ve_p 0.53/0.64/0.76/0.60 1.2/1.6/1.9/1.4 e

n_e 0.69 0.79

Te_e 0.63 0.88

TABLE I. Sample skewness and flatness of the time series sampled by the southwest / northwest / northeast / southeast MLP (Ie_s,Ve_f,Ve_p) and of the time series averaged over all electrodes (en_e,Te_e).

Compound quantities such as the local electric field and electron particle and heat fluxes are commonly estimated by combining floating potential and ion saturation current measurements. An estimator for the radial electric drift velocity is given by

U = V^S−V^N

B4_Z . (8)

Here B = 4.1 T gives the magnetic field at the probe head position, and (V^S−V^N)/4_Z denotes an estimator for the poloidal electric field. The north and south electrodes of the

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I_s/mA n_e/10¹⁸m⁻³ T_e/eV V_f/V V_p/V

Average 18 6.6 14 1.5 38

rms 8.7 1.8 2.7 5.5 8.2

TABLE II. Lowest order statistical moments of theIs,V_f,Vp,Te and ne data time series.

used probe head are vertically separated by4_Z = 2.2×10⁻³m. In the following we estimate the potential at either poloidal position as the average plasma potential asV^N/S = (Vp^NE/SE+ Vp^NW/SW)/2. Using the plasma potential instead of the floating potential to estimate the electric field includes effects of short-wavelength electron temperature perturbations on the radial velocity. Since toroidal plasma drifts may bias electrodes on the same magnetic flux surface to different electric potentials, the sample mean is subtracted from potential time series used in velocity estimators. Postulating that there is no stationary convection in the scrape-off layer we further subtract the moving average from radial velocity time series such that hUei_mv = 0.

With fast sampling of the electron density and temperature at hand, the radial electron particle and heat fluxes are estimated as

bΓ_n =ne_eU ,e (9)

b

Γ_T = hn_ei_mv

n_erms,mvTe_eUe+en_e hT_ei_mv

T_erms,mvUe +ne_eTe_eU .e (10) Here, en_e, Te_e, as well as moving average and moving root mean square time series denote quantities averaged over all four MLPs. This is done as to use all available data of the electron temperature as well as to average out outliers. Table II may be used to convert the amplitude of the estimator time series to physical units. We note that Eqs. ( 9) and 10 define fluctuation driven fluxes. The total fluctuation driven heat flux as defined above comprises a conductive contribution, a convective contribution, and a contribution driven by triple correlations.

IV. FLUCTUATION STATISTICS

Figure 3 shows PDFs of the rescaled ion saturation current, the electron density and the electron temperature time series. The two rightmost panels show the PDFs of the floating and the plasma potential. Least squares fits of the convolution of a normal and a Gamma

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−2.5 0.0 2.5 5.0 Ie_s 10⁻⁵

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

PDF

S= 1.20 F= 2.46

γ= 2.5

−2.5 0.0 2.5 5.0 e ne

S= 0.69 F= 0.79

γ= 8.4

−2.5 0.0 2.5 5.0 Te_e S= 0.63 F= 0.88

γ= 8.5

−5 0 5 Ve_f S=−0.83 F= 0.86

−5 0 5

Ve_p

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

S= 0.76 F= 1.92

FIG. 3. Probability distribution function of the rescaled ion saturation current, the electron density and temperature. Compared are least squares fits of the convolution of a Gamma and a normal distribution (black lines) to the PDFs. The two rightmost panels show the PDFs of the floating potential (red dots) compared to a normal distribution (black line) and of the plasma potential (purple dots).

distribution to the PDFs of Ie_s, en_e, and Te_e are shown by black lines. This distribution arises when assuming that the data time series are due to super-position of uncorrelated exponential pulses with an exponential amplitude distribution and additive white noise, see appendix A in [27]. The shape parameter of this distribution is given by γ =τ_d/τ_w, where τ_d is the pulse duration time and τ_w is the average pulse waiting time. Large values of γ describe time series that are characterized by significant pulse overlap. Realizations of the process described by Eq. ( A1) with small values of γ feature more isolated pulses. The signal to noise ratio of the additional white noise is given by 1/. For small values of the signal amplitude is governed by the arrival of exponential pulses, with additive noise contributing little to the signal amplitude.

The PDF of the ion saturation current time series features an elevated tail for large amplitude values. Sample coefficients of skewness and flatness are given by S = 1.2 and F = 2.5. A least-squares fit of the prediction by the stochastic model to the PDF yields γ = 2.5 and = 3.7×10⁻². This fit describes the PDF well over four decades in normalized probability. The PDFs of en_e and Te_e feature a similar shape but with less elevated tails for large, positive sample values. This is reflected in values of sample skewness and flatness given by S = 0.69 and F = 0.79 for en_e and by S = 0.63 and F = 0.88 for Te_e. Fitting the prediction by the stochastic model to the PDF of the sampled data yieldsγ = 8.4 and = 0

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for en_e and γ = 8.5 and = 0.13 for Te_e. Again, these parameters suggest a process with significant pulse overlap and little white noise.

Continuing with the PDF of the floating potential we find that negative sample values are more probable then positive sample values. This is reflected by negative value of the sample skewness, S =−0.83. The PDF deviates from a normal distribution, shown by the black line in the rightmost panel of Fig. 3, and reflected by a non-vanishing sample flatness F = 0.86. The PDF of the plasma potential features an elevated tail, similar to the PDF of Te_e. On the other hand negative Ve_p samples are more probable than negativeTe_e. Sample skewness and excess kurtosis are both non-vanishing for the plasma potential.

PDFs of Ie_s and Ve_f recorded by the other MLPs are qualitatively similar to those shown here. Interpreting the PDFs with the relationship given by Eq. ( 1) one may speculate that the elevated tail in the ion saturation current PDF is due to simultaneous large amplitude fluctuations of the electron density and temperature. This issue will be discussed further in the following sections.

Figure 4 shows the power spectral densities (PSDs) of the Ie_s,en_e,Ve_p, Ve_f and Te_e data time series. They all feature a similar shape, suggesting that fluctuations in the data time series are due to structures with similar characteristic time scales. Forf .3×10⁻³MHz the PSDs are flat before they roll over to approximately follow a power law, f⁻², for 3×10⁻²MHz. f . 0.1 MHz. For higher frequencies, the PSDs decay even more steep. A least squares fit of Eq. ( A5) to the data gives τ_d ≈ 15µs and λ ≈ 0 for all data time series. The black line gives the curve describe by Eq. ( A5) with just this pulse duration time and vanishing pulse rise time. Equation (A5) states that the flat part of the PSD as well as the roll-over frequency is determined by the pulse duration time τ_d. The pulse asymmetry parameter λ determines the slope of the PSD after the roll-over. We find that the prediction of the stochastic model with parameters found from least squares fits describe the experimental data well over approximately two decades.

Figure 5 shows the auto-correlation function for the data time series. The auto-correlation function of Ie_s, en_e, and Te_e decay approximately exponentially for τ . 20µs. The auto- correlation function of the Ve_f and Ve_p data time series decay faster than exponential. A least-squares fit of Eq. ( A4) to the data for τ < 25µs gives τ_d ≈ 15µs and λ ≈ 0 for Ie_s and Te_e. For ne_e a fit yields τ_d ≈ 16µs and a vanishing pulse asymmetry parameter. These parameters agree with the parameters estimated from fits to the PSDs of the data time

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10⁻³ 10⁻² 10⁻¹ 10⁰ f /MHz

10⁻² 10⁻¹ 10⁰ 10¹ 10²

PSD(f)

τ_d= 15µs, λ= 0

Ies:τd= 15.2µs e

ne:τd= 15.7µs Tee:τd= 11.5µs Vef :τd= 12.3µs Vep:τd= 12.4µs

FIG. 4. Power spectral density of the ion saturation current, the floating and the plasma potential, and the electron density and temperature. The black line denotes Eq. ( A5) withτ_d = 15µs.

0 10 20 30 40 50

τ /µs 0.2

0.4 0.6 0.8 1.0

R(τ)

τd= 15.0µs, λ= 0

Ies

e ne

Tee

Vef

Vep

FIG. 5. Auto-correlation of the rescaled ion saturation current, electron density and temperature as well as the plasma and the floating potential. Compared is Eq. ( A4) withτ_d= 15µs andλ= 0, denoted by the black line.

series.

Figure 6 shows cross-correlation functions between the ion saturation current and the other data time series. The correlation functions R_I_e

s,ene(τ) and R_I_e

s,Tee(τ) appear similar in shape. They feature maximum correlation amplitudes of approximately 0.75 at nearly vanishing time lag and are slightly asymmetric, with the correlation amplitude decaying slower for positive than for negative time lags. The cross-correlation function for the plasma potential, R

Ies,Vep(τ), features a maximal correlation amplitude of approximately 0.6 at vanishing time lag. It decays slower to zero for positive time lags than for negative time lags. The cross-correlation function for the floating potential, R

Ies,eV_f(τ) presents a minimal correlation

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−25 0 25 τ /µs

−0.25 0.00 0.25 0.50 0.75 1.00

R(τ)

Ies

e ne

Tee

Ve_f Vep

FIG. 6. Cross-correlation between all data time series and the ion saturation current.

amplitude of approximately−0.4 atτ ≈2µs. It appears symmetric around the minimum for

−5µs.τ .8µs but decays faster to zero for τ >0 than forτ < 0 for large lags. Observing that all auto-correlation functions vanish for time lags greater than 50µs we note that we do not observe any long-range correlations.

Complementary to the auto-correlation function we proceed by studying the time series using the conditional averaging method [86]. The conditionally averaged waveform of a signal Φ is computed by averaging sub-records centered around local maxima of a reference signal Ψ which exceed a threshold value, typically taken to be 2.5 times the time series root mean square value:

C_Φ,_e_Ψ_e(τ) =hΦ(τ)|e Ψ(τe = 0)>2.5, Ψe⁰(0) = 0i. (11) Here the prime denotes a derivative. To ensure that the conditionally averaged waveform is computed from independent samples, the local maxima are required to be separated by the same interval length on which Eq. ( 11) is computed. For the data sets at hand we choose

−25µs≤τ ≤25µs.

Figure 7 shows the conditionally averaged waveform of the Ie_s,ne_e,Te_e,Ve_f andVe_p data time series, usingIe_sas a reference signal. Approximately 4000 maxima are detected in theIe_stime series. The conditionally averaged waveform of the ion saturation current is strongly peaked and decays faster then exponentially to zero for large time lags. The average amplitude of the local ion saturation current maxima is approximately three times the time series root mean square value. The conditionally averaged waveforms of the electron density and temperature are both well approximated by a two-sided exponential function. The maxima

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−20 −10 0 10 20 τ /µs

−1 0 1 2 3

hΦ(τ)|eIs(0)>2.5i

Ies

e ne

Tee

Vef

Vep

FIG. 7. Conditionally averaged waveforms of the data time series, centered around large amplitude maxima in the ion saturation current time series. Dashed lines show least squares fits of a two-sided exponential waveform, given by Eq. ( A2), to the conditionally averaged waveforms.

of their waveforms are approximately two times the root mean square value of their respective time series. The conditionally averaged waveform of the Ve_p time series appears triangular, with the maxima in phase with local maxima of the Ie_s time series. The conditionally averaged waveform of the floating potential presents a negative peak with an amplitude of approximately −1, occurring at τ ≈ −2µs. Compared to the averaged waveforms are least square fits of a two-sided exponential waveform, given by Eq. ( A2), to the data, marked by dashed lines in Fig. 7. Table III lists their fit parameters. The average waveform duration time is between 12 and 16µs, comparable to τ_d estimated by fits to the auto-correlation function and power spectral densities of the signals. The pulse asymmetry parameter for all fits is given by approximately 0.4.

Waveform hIe_s|eI_s(0)>2.5i hne_e|eI_s(0)>2.5i hTe_e|eI_s(0)>2.5i

τ_d/µs 11 16 16

λ=τ_r/τ_d 0.37 0.37 0.40

TABLE III. Duration time of the last squares fits shown in Fig. 7 and the waveform asymmetry parameterλ.

Computing the time lag between successive, conditional maxima of the time series yields the waiting time statistics for large-amplitude events. Figure 8 shows the waiting time PDF of the Ie_s, en_e and Te_e time series. Compared are PDFs of exponentially distributed variables. Their scale parameter is given by a maximum likelihood estimate of the waiting

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0.25 0.50 0.75 1.00 1.25 1.50 τw/ms

10⁻⁵ 10⁻⁴ 10⁻³

PDF(τw)

Ies:hτwi= 0.20ms e

ne:hτwi= 0.28ms Tee:hτwi= 0.25ms

FIG. 8. PDFs of the waiting times between large-amplitude bursts in then_e,T_e, andI_stime series, recorded by the southwest MLP. The full lines show PDFs of a truncated exponential distribution with a scale parameter given by a maximum likelihood estimate to the data points.

time distribution. The resulting distributions describe the data well over approximately two decades in probability. Average waiting times are given by approximately 0.25 ms for Te_e and 0.28 ms for en_e. The average waiting time between large-amplitude bursts in Ie_s is approximately 0.20 ms. We note that the exact numerical values depend slightly on the threshold value and the conditional window length used.

The PDFs of the signals local maxima are shown in Fig. 9. As for the average waiting times, the PDFs are well described by a truncated exponential distribution. The scale parameter, found by maximum likelihood estimates of the data, are given by hAi ≈ 1.0 for Ie_s, hAi ≈ 0.77 for en_e, and by hAi ≈ 0.83 for Te_e. Given the threshold amplitude of 2.5, this translates to an average burst amplitude of the rescaled signals between 3.3 and 3.5 times the root-mean-square value of the data time series, consistent with the amplitude of the conditionally averaged waveforms shown in Fig. 9.

V. RADIAL VELOCITY AND FLUXES

In the following the statistical properties of the radial velocity and electron particle and heat fluxes are discussed. Figure 10 shows 1 ms long time series of the estimators given by Eqs. ( 8) - (10), computed on the same time interval as the time series shown in Fig. 1. The full (dashed) line in the upper panel denotes the radial velocity estimated fromV_p (V_f) samples. In the middle panel the full (dashed) line denotes the radial electron

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3 4 5 6 7 A

10⁻² 10⁻¹ 10⁰

PDF(A)

Ies:hAi= 1.01 e

ne:hAi= 0.77 Tee:hAi= 0.83

FIG. 9. PDF of large amplitude local maxima in the rescaled time series. Full lines show the PDFs of a truncated exponential distribution with a scale parameter given by a maximum likelihood estimate to the data time series.

0.00 0.25 0.50 0.75 1.00

−5 0 5

−5 0

5 rms = 1.25 mean = 0.42

0.00 0.25 0.50 0.75 1.00

time/ms 0

100

rms = 15.7 mean = 4.54

FIG. 10. Estimators for the radial velocity (upper panel) and the radial electron particle, and heat flux (middle and lower panel). The full (dashed) line in the upper panel denotes the radial velocity estimated from V_p (V_f) samples. The full (dashed) line in the middle panel denotes the radial electron flux esimated fromne_e and Ue_V_p (Ie_s and Ue_V_f) samples. The time interval is identical to the one presented in Fig. 1.

flux estimated from n_e and V_p (I_s and V_f) samples. The radial velocity time series show fluctuations on a similar time scale as seen for the time series shown in Fig. 1. There is no qualitative difference between Ue estimated from the floating potential and from the plasma potential. Both positive and negative local maxima appear with nearly equal frequency, not exceeding 5 in normalized units. The bΓ_n time series feature predominantly positive fluctuation amplitudes on a similar temporal scale as theUe time series. Using ion saturation

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−5.0 −2.5 0.0 2.5 5.0 Ue

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

PDF(eU)

UeV_f : S=−0.15, F= 0.47 Ue_V_p:S= 0.36, F= 2.39

Ue_V_f UeVp

γ= 10

FIG. 11. PDF of the rescaled radial electric drift velocity. Compared is the PDF predicted from the stochastic model, given by Eq.(A9) [46], for a Laplace distribution of pulse amplitudes.

current and floating potential to estimate the particle flux yields almost indistinguishable estimator samples. The sample mean and root mean square value are given by 0.42 (0.48) and 1.25 (1.20) respectively, using n_e and V_p (I_s and V_f) samples. The radial heat flux time seriesΓb_T features large-amplitude bursts exceeding 80 in normalized units. Large-amplitude temperature fluctuations, which appear in phase with large-amplitude particle flux events, give rise to this large fluctuation level. Only few large, negative heat flux events are recorded.

Figure 11 presents the PDF of the radial velocity estimator given by Eq. ( 8). The PDF of Ue_V_e

f appears symmetric with exponential tails for both positive and negative sample values, compatible with S = −0.15 and F = 0.47. The PDF of Ue_V_e

p is almost identical to the PDF computed using floating potential measurements, but notably features an elevated tail for large amplitude samples Ue_V_e_p & 2.5. A correlation analysis of samples V_f^S −V_f^N and T_e^S−T_e^N showed no correlation between large-amplitude potential differences to large amplitude electron temperature differences, which may have explained this artifact in the PDF. The coefficient of sample skewness for Ue_V_e

f is slightly negative, while the elevated tail of PDF(Ue

Vep) yields a slightly positive coefficient of sample skewness. Compared to the PDF is the probability distribution function of the process defined by Eq. ( A1) with Laplace distributed pulse amplitudes, which allows for positive as well as negative pulse amplitudes [46 and 87]. Estimatingγ by a least squares fit to the PDF ofUe

Ve_f yieldsγ ≈10. This value is comparable with the intermittency parameter for the en_e and Te_e data time series and is larger by a factor of approximately 4 than for the Ie_s time series.

The auto-conditionally averaged waveform of large-amplitude velocity fluctuations, com-

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−20 −10 0 10 20 τ /µs

−2 0 2

heU(τ)||eU(0)|>2.5i

Ue_V_f UeVp

FIG. 12. Conditionally averaged waveform of local extrema in the radial velocity time series.

puted from approximately 3000 events, are shown in Fig. 12. The average waveform is approximately triangular for the Ue

Vep time series while it is less peaked for the Ue

Vef time series. The duration time of both waveforms is approximately 5µs, smaller by a factor of three than the conditionally averaged waveforms of the electron density and temperature.

The PDF of the waiting times between local extrema in the Ue time series, both positive and negative, are shown in Fig. 13. Compared are PDFs of exponentially distributed variables with scale parameters given by hτ_wi= 0.08 ms for Ue

Vep and by hτ_wi = 0.13 ms for Ue

Ve_f. These parameters have been estimated by a maximum likelihood estimate of the respective waiting time data. The resulting distributions describes the waiting times well over approximately two decades in probability. Varying the minimum separation between detected local extrema changes the average waiting time only little since positive and negative maxima are detected independently of each other.

The PDF of the local extrema is shown in Fig. 14. Compared to the positive and negative legs of the distribution are truncated exponential distributions for |A| > 2.5. Each distribution has been multiplied by 1/2 to normalize the integral of both PDFs to unity. A least squares fit to the Ue

Vep data yields a scale parameter of approximately 1 for both positive and negative amplitudes. Together with the result that the waiting times between large local maxima of the velocity time series are well described by an exponential distribution, these findings corroborate the hypothesis to interpret the radial velocity time series as a super-position of uncorrelated pulses described by Eq. ( A1).

Figure 15 presents the PDF of the radial particle fluxes computed using eithern_e andV_p samples or I_s and V_f samples. The PDFs are almost indistinguishable. They are strongly

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0.1 0.2 0.3 0.4 0.5 0.6 τw/ms

10⁻⁴ 10⁻³ 10⁻²

PDF(τw)

hτwi= 0.13 ms hτwi= 0.08 ms

FIG. 13. PDF of the waiting times between successive, positive or negative, extrema in the radial velocity time series. Square (circle) plot markers denote the radial velocity estimated fromVp (V_f) samples. Compared are exponential distributions with a scale parameter given by a maximum likelihood estimate of the waiting times.

−10 −5 0 5 10

A 10⁻³

10⁻² 10⁻¹ 10⁰

PDF(A)

FIG. 14. PDF of the local maxima in the radial velocity time series. Square (circle) plot markers denote the radial velocity estimated from V_p (V_f) samples. Compared are fits on exponential distribution multiplied by a factor 1/2, forA >2.5 andA <2.5.

peaked at zero and feature non-exponential tails for both positive and negative sample values. Positive sample values have a much higher probability than negative sample values.

This is reflected by coefficients of sample skewness and excess kurtosis given by S = 4.3 (3.8) and F = 65 (33), respectively.

The PDF of the radial heat flux, shown by circles in Fig. 16, presents a similar shape with heavy tails for large sample values. Sample coefficients of skewness and flatness are given by S = 7.3 and F = 190. Also shown are PDFs of the conductive heat flux

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−5 0 5 10 15 Γbn

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

PDF(bΓn)

Γb_n,_n_e_e:S= 4.3F= 65 Γb_n,_I_e_s :S= 3.8F = 33

FIG. 15. PDF of the normalized radial electron flux. Square (circle) plot markers denote the radial electron flux estimated using ene and UVp (Ies and UV_f) samples.

−40 −20 0 20 40 60 80 ΓbT

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

PDF(bΓT)

S= 7.3, F= 190 S= 5.3, F= 170 S= 4.3, F= 64 S= 18, F= 960

FIG. 16. PDF of the total radial heat flux (circle), the conductive (triangle left), convective heat fluxes (triangle right), and the heat flux due to triple correlations(cross).

(hn_ei_mv/n_erms,mv)Te_eUe (triangle left), the convective heat flux, ne_e(hT_ei_mv/T_erms,mv)Ue (triangle right), and triple correlations en_eTe_eUe (cross). PDFs of the conductive and convective heat fluxes appear similar in shape as the total heat flux. However, large-amplitude convective heat flux samples occur more frequently than conductive heat flux samples of equal magnitude. The PDF of the heat flux due to triple correlations is strongly peaked for small amplitudes and skewed towards positive sample values.

The sample averages and root mean square values of the various contributions to the total heat flux are listed in Tab. IV. This data shows that 38% of the total fluctuation driven heat flux is due to conduction, 56% due to convection, and 6% due to triple correlations.

For both the particle and the total heat flux we find that their root mean square value is

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Γ_n Γ_T hn_ei_mvTe_eUe hT_ei_mven_eUe en_eTe_eUe Average 10²⁰m⁻²s⁻¹ 2.9 70 eV 27 eV 39 eV 4.2 eV Root mean square 10²⁰m⁻²s⁻¹ 8.8 250 eV 95 eV 120 eV 55 eV

TABLE IV. Sample average and root mean square value of the contributions to the radial fluxes.

−2.5 0.0 2.5 5.0 e

ne

−5.0

−2.5 0.0 2.5 5.0

eU

R= 0.41 10⁻³

10⁻ 10−1 2

10⁻⁴ 2 5 10⁻³ 2 5 10⁻² 2 5 10⁻¹ 2 5

PDF

FIG. 17. Joint PDF of the radial velocity and the electron density fluctuations.

approximately two-three times their mean value. This also holds for the conductive and the convective heat fluxes. The relative fluctuation level of the heat flux due to triple correlations is approximately 12.

We continue by discussing the correlations between the electron density and temperature and the radial velocity fluctuation time series. Figure 17 presents the joint PDF of the fluctuating radial velocity and electron density. The linear sample correlation coefficient is given by R = 0.41, consistent with the slightly tilted shape of the ellipsoids capturing probabilities less than 10⁻¹, 10⁻² and 10⁻³. Large-amplitude fluctuations are enclosed by equi-probability ellipsoids whose semi-minor axis increases with decreasing probability. Neg- ative large-amplitude velocity fluctuations, Ue . −2.5, appear in phase with small positive and negative density fluctuations. Positive, large amplitude density fluctuations, en_e & 2.5, appear on average in phase with positive velocity fluctuations. Negative density fluctuations appear on average with vanishing velocity fluctuations while positive velocity fluctuations,

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−2.5 0.0 2.5 5.0 Tee

−5.0

−2.5 0.0 2.5 5.0

eU

R= 0.29 10⁻

3

10⁻ 10−1 2

10⁻⁴ 2 5 10⁻³ 2 5 10⁻² 2 5 10⁻¹ 2 5

PDF

FIG. 18. Joint probability distribution function of the radial velocity and electron temperature fluctuations.

Ue &2.5 appear on average in phase with positive density fluctuations.

The joint PDF of the fluctuating radial velocity and the electron temperature, shown in Fig. 18, features some qualitative similarities to the joint PDF of the velocity and density fluctuations. Small-amplitude fluctuations are correlated, captured by a tilted ellipsoid for a joint probability approximately less than 0.1. The sample correlation coefficient for the time series is given by R = 0.29. Large-amplitude fluctuations are captured by equi-probability contours whose shape increasingly deviates from an ellipse with decreasing probability. Es- pecially are large, negative velocity fluctuations observed which are in phase with small, positive temperature fluctuations. Large negative temperature fluctuations are in phase with small velocity fluctuations, Ue ≈ 0, with less scatter than observed for the density fluctuations. Large positive temperature fluctuations are on average in phase with positive temperature fluctuations, also with larger scatter than observed for the density fluctuations. Large velocity fluctuations with Ue & 2.5 are correlated with positive temperature fluctuations. Similar to the correlation to density fluctuations, are large negative velocity fluctuations, Ue .2.5 on average in phase with small, positive temperature fluctuations.

Figure 19 presents the radial velocity amplitudes encoded in a scatter plot of the electron

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−2.5 0.0 2.5 5.0 e

ne

−2.5 0.0 2.5 5.0

eTe

−6

−4

−2 0 2 4 6

Ue

FIG. 19. Amplitude of the velocity fluctuations as a function of the electron density and temperature fluctuation amplitude.

density and temperature fluctuations. Large-amplitude velocity fluctuations are in phase with large-amplitude fluctuations in both electron temperature and density. The magnitude of Ue increases with the amplitude of en_e and Te_e. Few negative velocity fluctuations are observed for ne_e& 0 and Te_e &0. Negative velocity fluctuations are observed for en_e&0 and Te_e .0, as well as for Te_e&0 and en_e.0.

We continue by investigating how these fluctuations contribute to the radial heat flux. For this we present the conditionally averaged wave forms of the fluctuating time series, centered around heat flux events exceeding 25 in normalized units, shown in Fig. 20. Here we use the conductive and convective heat flux, as well as contributions from triple correlations as reference signals. The left panels show the conditionally averaged waveforms of the respective heat fluxes and the right panels show their conditional variance (CV) [88]. The conditional variance describes the average deviation of the individual waveforms from the average waveform. A value of CV = 0 describes identical individual waveforms while a value of CV = 1 describes random individual waveforms. In total 2692 local maxima are identified in the conductive heat flux time series, 3963 in the convective heat flux time series and 992 in the triple correlations time series. These counts agree with the PDFs of individual heat